1 / 55

Recent developments in the type IIB matrix model

Recent developments in the type IIB matrix model. Jun Nishimura (KEK & SOKENDAI) 8-15 September, 2013 “Workshop on Noncommutative Field Theory and Gravity” Corfu, Greece. 1. Introduction. Particle physics. Cosmology. successful phenomenological models.

ziv
Download Presentation

Recent developments in the type IIB matrix model

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Recent developments in the type IIB matrix model Jun Nishimura (KEK & SOKENDAI) 8-15 September, 2013 “Workshop on Noncommutative Field Theory and Gravity” Corfu, Greece

  2. 1. Introduction Particle physics Cosmology successful phenomenological models Standard Model Inflation and Big Bang CMB (WMAP, PLANCK) structure formation of galaxies nucleosynthesis discovery of SM-like Higgs particle at LHC experiments and observations What is the origin of Higgs particle ? SM fermions ? (3 generations) the SM gauge group ? What is the origin of inflaton , its potential, its initial condition ? dark matter, dark energy ? missing fundamental understanding serious naturalness problems cosmological constant problem hierarchy problem All these problems may be solved by superstring theory ! string phenomenology string cosmology

  3. The most fundamental issue in superstring theory is space-time dimensionality (10-dimensional space-time is required for consistency.) conventional approach: • compactify extra dimensions. infinitely many perturbative vacua space-time dimensionality gauge symmetry matter (#generations) with various • D-branes (representing certain nonperturbative effects) enriched both string phenomenology and string cosmology intersecting D-brane models D-brane inflation models flux compactification … Too many models, and no predictive power !!!

  4. However, we should not forget that all these perspectives are obtained from mostly perturbative studies of superstring theory including at most the nonpertubative effects represented by the existence of D-branes. A totally new perspective might appear if one studies superstring theory in a complete nonperturbative framework. c.f.) lattice gauge theory in the case of QCD confinement of quarks, hadron mass spectrum, etc. (Can never be understood from perturbation theory !!!)

  5. type IIB matrix model (Ishibashi, Kawai, Kitazawa, Tsuchiya, 1996) nonperturbative formulation of superstring theory based on type IIB superstring theory in 10d M • The connection to perturbative formulations can be seen manifestly • by considering 10d type IIB superstring theory in 10d. IIA Het E8 x E8 Het SO(32) IIB worldsheet action, light-cone string field Hamiltonian, etc. I • A natural extension of the “one-matrix model”, which is established as • a nonperturbative formulation of non-critical strings regarding Feynman diagrams in matrix models as string worldsheets. • It is expected to be a nonperturbative formulation of • the unique theory underlying the web of string dualities. (Other types of superstring theory can be represented as perturbative vacua of type IIB matrix model)

  6. Hermitian matrices type IIB matrix model SO(9,1) symmetry Lorentzian metric is used to raise and lower indices Wick rotation Euclidean matrix modelSO(10) symmetry

  7. Lorenzian v.s. Euclidean There was a reason why no one dared to study the Lorentzian model for 15 years! opposite sign! extremely unstable system. Once one Euclideanizes it, positive definite! flat direction ()is lifted up due to quantum effects. Aoki-Iso-Kawai-Kitazawa-Tada (’99) Euclidean model is well defined without the need for cutoffs. Krauth-Nicolai-Staudacher (’98), Austing-Wheater (’01)

  8. Recent developments • Euclidean model • Interpretation of these results is unclear, though. • Wick rotation is not justifiable unlike in ordinary QFT! • Lorentzian model • Can be made well-defined by introducing IR cutoffs • and removing them in the large-N limit. • Real-time evolution can be extracted from matrix configurations. • Expanding 3d out of 9d (SSB), inflation, big bang,… • A natural solution to the cosmological constant problem. • Realization of the Standard Model (Tsuchiya’s talk) Lorentzian version of type IIB matrix model is indeed the correct nonperturbative formulation of superstring theory, which describes our Universe.

  9. Plan of the talk • Introduction • Euclidean type IIB matrix model • Lorentzian type IIB matrix model • Expanding 3d out of 9d • Exponential/power-law expansion • Time-evolution at much later times • Summary and future prospects

  10. 2. Euclidean type IIB matrix model J.N.-Okubo-Sugino, JHEP1110(2011)135, arXiv:1108.1293 Aoyama-J.N.-Okubo, Prog.Theor.Phys. 125 (2011) 537, arXiv:1007.0883 Anagnostopoulos -Azuma-J.N., arXiv:1306.6135

  11. Most recent results based on the Gaussian expansion method J.N.-Okubo-Sugino, JHEP1110(2011)135, arXiv:1108.1293 KNS extended direction shrunken direction d=3 gives the minimum free energy universal shrunken direction Space-time has finite extent in all directions. ＳＳＢ ＳＯ（10）　　　　　ＳＯ（3） This is an interesting dynamical property of the Euclideanized type IIB matrix model. However, its physical meaning is unclear…

  12. Results for 6d SUSY model (Gaussian expansion method) Aoyama-J.N.-Okubo, Prog.Theor.Phys. 125 (2011) 537, arXiv:1007.0883 extended direction shrunken direction : KNS d=3 gives the minimum free energy universal shrunken direction ＳＳＢ ＳＯ(6) 　　　　ＳＯ（3） The mechanism for the SSB is demonstrated by Monte Carlo studies in this case. “phase of the fermion determinant”

  13. No SSB if the phase is omitted Anagnostopoulos –Azuma-J.N., arXiv:1306.6135 eigenvalues : All the eigenvalues converge to the same value at large N !

  14. Effects of the phase : universal shrunken direction Anagnostopoulos -Azuma-J.N., arXiv:1306.6135 SO(2) vacuum SO(3) vacuum effects of the phase of fermion determinant analysis based on the factorization method SO(4) vacuum SO(5) vacuum Anagnostopoulos –J.N. Phys.Rev. D66 (2002) 106008 hep-th/0108041 Consistent with the GEM calculations ! c.f.)

  15. After all, the problem was in the Euclideanization ? • In QFT, it can be fully justified as an analytic continuation. • 　　　　（That’s why we can use lattice gauge theory.) • On the other hand, it is subtle in gravitating theory. • (although it might be OK at the classical level…) • Quantum gravity based on dynamical triangulation (Ambjorn et al. 2005） • (Problems with Euclidean gravity may be overcome in Lorentzian gravity.) • Coleman’s worm hole scenario for the cosmological constant problem • (A physical interpretation is possible only by considering the Lorentzian • version instead of the original Euclidean version.) Okada-Kawai (2011) • Euclidean theory is useless for studying the real time dynamics • such as the expanding Universe.

  16. 3. Lorentzian type IIB matrix model Kim-J.N.-TsuchiyaPRL 108 (2012) 011601[arXiv:1108.1540]

  17. Definition of Lorentzian type IIB matrix model Kim-J.N.-TsuchiyaPRL 108 (2012) 011601[arXiv:1108.1540] partition function This seems to be natural from the connection to the worldsheet theory. (The worldsheet coordinates should also be Wick-rotated.)

  18. Regularization and the large-N limit Unlike the Euclidean model, the Lorentzian model is NOT well defined as it is. • The extent in temporal and spatial directions should be made finite. • (by introducing cutoffs) In what follows, we set without loss of generality. • It turned out that these two cutoffs can be removed in the large-N limit. • (highly nontrivial dynamical property) • Both SO(9,1) symmetry and supersymmetry are broken explicitly by the cutoffs. • The effect of this explicit breaking is expected to disappear • in the large-N limit.（needs to be verified.)

  19. Sign problem can be avoided ! (2) (1) The two possible sources of the problem. (1)Pfaffian coming from integrating out fermions The configurations with positive Pfaffian become dominant at large N. In Euclidean model, This complex phase induces the SSB of SO(10) symmetry. J.N.-Vernizzi (’00), Anagnostopoulos-J.N.(’02)

  20. Sign problem can be avoided ! (Con’d) (2) What shall we do with The same problem occurs in QFT in Minkowski space (Studying real-time dynamics in QFT is a notoriously difficult problem.) First, do homogenous in

  21. 4. Expanding 3d out of 9d Kim-J.N.-TsuchiyaPRL 108 (2012) 011601[arXiv:1108.1540]

  22. How to extract time-evolution SU(N) transformation definition of time “t” small non-trivial dynamical property small

  23. Band-diagonal structure We choose as the block size

  24. Spontaneous breaking of SO(9) Kim-J.N.-Tsuchiya,PRL 108 (2012) 011601 SSB “critical time”

  25. 5. Exponetial/power-law expansion Ito-Kim-Koizuka-J.N.-Tsuchiya, in prep. Ito-Kim-J.N.-Tsuchiya, work in progress

  26. Exponentialexpansion Ito-Kim-J.N.-Tsuchiya, work in progress fitted well to SO(9) SO(3) Exponential expansion Inflation

  27. Effects of fermionic action dominant term at early times dominant term at late times keep only the first term simplified model at early times simplified model at late time quench fermions

  28. Exponential expansion at early times Ito-Kim-Koizuka-J.N.-Tsuchiya, in prep. • simplified model at early times exponential expansion The first term is important for exponential expansion.

  29. Power-law expansion at late times Ito-Kim-J.N.-Tsuchiya, work in progress • simplified model at late times Radiation dominated FRW universe

  30. Expected scenario for the full Lorentzian IIB matrix model radiation dominated FRW universe E-folding is determined by the dynamics ! inflation （big bang）

  31. 6. Time-evolution at much later times S.-W. Kim, J. N. and A.Tsuchiya, Phys. Rev. D86 (2012) 027901 [arXiv:1110.4803] S.-W. Kim, J. N. and A.Tsuchiya, JHEP 10 (2012) 147 [arXiv:1208.0711]

  32. Time-evolution at much later times The cosmic expansion makes each term in the action larger at much later times. Classical approximation becomes valid. • There are infinitely many classical solutions. (Landscape) • There is a simple solution representing a (3+1)D expanding universe, • which naturally solves the cosmological constant problem. • Since the weight for each solution is well-defined, • one should be able to determine the unique solution that dominates • at late times. • By studying the fluctuation around the solution, • one should be able to derive the effective QFT below the Planck scale. J. N. and A.Tsuchiya, PTEP 2013 (2013) 043B03 [arXiv:1208.4910]

  33. General prescription for solving EOM • variational function • classical equations of motion • commutation relations Eq. of motion & Jacobi identity Can be made finite- dimensional by imposing simplifying Ansatz. Lie algebra Unitary representation classical solution

  34. An example of SO(4) symmetric solution (RxS3space-time) EOM algebra uniformly distributed on a unit S3 Space-space is commutative.

  35. An example of SO(4) symmetric solution (RxS3space-time) cont’d primary unitary series Block size can be taken to be n=3. the extent of space space-time noncommutativity commutative space-time ! cont. lim. consistent!

  36. An example of SO(4) symmetric solution (RxS3space-time) cont’d This part can be identified as a viable late-time behavior. cosmological const. explains the accelerated expansion at present time Cosmological constant disappears in the far future. A natural solution to the cosmological constant problem.

  37. 7. Summary and future prospects

  38. type IIB matrix model (1996) Summary A nonperturbative formulation of superstring theory based on type IIB theory in 10d. The problems with the Euclidean model have become clear. Lorentzian model : untouched until recently because of its instability Monte Carlo simulation has revealed its surprising properties. • Awell-defined theory can be obtained by introducing cutoffs • and removing them in the large-N limit. • The notion of “time evolution” emerges dynamically When we diagonalize , has band-diagonal structure. • After some “critical time”, the space undergoes the SSB of SO(9), • and only 3 directions start to expand. • Exponential expansion observed (Inflation, no initial condition problem.) • Power-law ( ) expansion observed in a simplified model for later times. • Classical analysis is valid for much later times. • A natural solution to the cosmological coonstant problem suggested.

  39. Future prospects • Observe directly the transition from the exponential expansion • to the power-law expansion by Monte Carlo simulation. • Does the transition to commutative space-time • (suggested by a classical solution)occur at the same time ? • Can we calculate the density fluctuationto be compared with CMB ? • Can we read off the effective QFT below the Planck scale from fluctuations • around a classical solution ? • Does Standard Model appear at low energy ? (Tsuchiya’s talk) Various fundamental questions in particle physics and cosmology : the mechanism of inflation, the initial value problem, the cosmological constant problem, the hierarchy problem, dark matter, dark energy, baryogenesis, the origin of the Higgs field, the number of generations etc.. It should be possible to understand all these problems in a unified manner by using the nonperturbative formulation of superstring theory.

  40. Backup slides

  41. Previous works in the Euclidean matrix model A model with SO(10) rotational symmetry instead of SO(9,1) Lorentz symmetry Dynamical generation of 4d space-time ? SSB of SO(10) rotational symmetry • perturbative expansion around diagonal configurations, • branched-polymer picture • Aoki-Iso-Kawai-Kitazawa-Tada(1999) • The effect of complex phase of the fermion determinant (Pfaffian) • J.N.-Vernizzi (2000) • Monte Carlo simulation • Ambjorn-Anagnostopoulos-Bietenholz-Hotta-J.N.(2000) • Anagnostopoulos-J.N.(2002) • Gaussian expansion method • J.N.-Sugino (2002)、Kawai-Kawamoto-Kuroki-Matsuo-Shinohara(2002) • fuzzy • Imai-Kitazawa-Takayama-Tomino(2003)

  42. Emeregence of the notion of “time-evolution” mean value small band-diagonal structure represents the state at the time t small

  43. The emergence of “time” Supersymmetry plays a crucial role! Calculate the effective action for at one loop. contributes contributes Contribution from van der Monde determinant Altogether, Zero, in a supersymmetric model ! Attractive force between the eigenvalues in the bosonic model, cancelled in supersymmetric models.

  44. The time-evolution of the extent of space symmetric under We only show the region

  45. SSB of SO(9) rotational symmetry SSB “critical time”

  46. What can we expect by studying the time-evolution at later times • What is seen by Monte Carlo simulation so far is: • the birth of our Universe • What has been thought to be the most difficult • from the bottom-up point of view, can be studied first. • This is a typical situation in a top-down approach ! • We need to study the time-evolution at later times • in order to see the Universe as we know it now! • Does inflation and the Big Bangoccurs ? • (First-principles description based on superstring theory, instead of just a • phenomenological description using “inflaton”; comparison with CMB etc.. • How does the commutative space-timeappear ? • What kind of massless fields appear on it ? • accelerated expansion of the present Universe (dark energy), • understanding the cosmological constant problem • prediction for the end of the Universe(Big Crunch or Big Rip or...)

  47. Ansatz extra dimension is small (compared with Planck scale) commutative space

  48. Simplification Lie algebra e.g.)

  49. d=1 case SO(9) rotation Take a direct sum distributed on a unit S3 (3+1)D space-time R×S3 A complete classification of d=1 solutions has been done. Below we only discuss a physically interesting solution.

  50. SL(2,R) solution • SL(2,R) solution • realization of the SL(2,R) algebra on

More Related